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CROSS CURVATURE FLOW ON A NEGATIVELY CURVED SOLID TORUS JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG Abstract. The classic 2π-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the “2π-metric” and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds, and integral convergence to hyperbolic for the metrics under consideration. 1. Introduction In this note, we outline a program that uses cross curvature flow to answer certain questions inspired by the 2π-Theorem of Gromov and Thurston. We begin and make partial progress toward completing this program. More specifically, we consider cross curvature flow on a class of negatively curved metrics on the solid torus, the simplest nontrivial handlebody. This is motivated by the “Dehn surgery” construction in 3-manifold topology, of which we give a brief account below. We apply the flow to a negatively curved metric described by Gromov–Thurston on a solid torus with prescribed Dirichlet boundary conditions. Perelman’s use of Ricci flow with surgery to prove the Geometrization Conjec- ture demonstrates the considerable power of geometric flows to address questions about 3-manifolds [24, 25]. Subsequent work, that of Agol–Storm–Thurston [3], for example, shows that Ricci flow may give information even about 3-manifolds known a priori to admit a metric of constant curvature. The results of [3] are obtained by using the monotonicity of volume under Ricci flow with surgery. Ricci flow is nonetheless an imperfect tool for analyzing hyperbolic 3-manifolds — those admitting metrics with constant negative sectional curvatures. According to Geometrization, such manifolds form the largest class of prime 3-manifolds. Paradoxically, they are also the least understood class. In particular, the facts that Ricci flow does not preserve negative curvature and that singularities must be resolved by surgery leaves certain basic questions unanswered. For example, one may ask whether the space of negatively curved metrics on a manifold M is connected or contractible, or analogous questions for the space of metrics with negative curvature satisfying various pinching conditions. In higher dimensions, these questions are known to have negative answers [10, 11, 12]; but it has been conjectured and seems plausible that their answers should be positive if M is a hyperbolic 3-manifold. D.K. acknowledges NSF support in the form of grants DMS-0505920 and DMS-0545984. 1

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CROSS CURVATURE FLOW ON ANEGATIVELY CURVED SOLID TORUS

JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

Abstract. The classic 2π-Theorem of Gromov and Thurston constructs a

negatively curved metric on certain 3-manifolds obtained by Dehn filling. ByGeometrization, any such manifold admits a hyperbolic metric. We outline

a program using cross curvature flow to construct a smooth one-parameter

family of metrics between the “2π-metric” and the hyperbolic metric. We makepartial progress in the program, proving long-time existence, preservation of

negative sectional curvature, curvature bounds, and integral convergence to

hyperbolic for the metrics under consideration.

1. Introduction

In this note, we outline a program that uses cross curvature flow to answercertain questions inspired by the 2π-Theorem of Gromov and Thurston. We beginand make partial progress toward completing this program. More specifically, weconsider cross curvature flow on a class of negatively curved metrics on the solidtorus, the simplest nontrivial handlebody. This is motivated by the “Dehn surgery”construction in 3-manifold topology, of which we give a brief account below. Weapply the flow to a negatively curved metric described by Gromov–Thurston on asolid torus with prescribed Dirichlet boundary conditions.

Perelman’s use of Ricci flow with surgery to prove the Geometrization Conjec-ture demonstrates the considerable power of geometric flows to address questionsabout 3-manifolds [24, 25]. Subsequent work, that of Agol–Storm–Thurston [3], forexample, shows that Ricci flow may give information even about 3-manifolds knowna priori to admit a metric of constant curvature. The results of [3] are obtained byusing the monotonicity of volume under Ricci flow with surgery.

Ricci flow is nonetheless an imperfect tool for analyzing hyperbolic 3-manifolds— those admitting metrics with constant negative sectional curvatures. Accordingto Geometrization, such manifolds form the largest class of prime 3-manifolds.Paradoxically, they are also the least understood class. In particular, the factsthat Ricci flow does not preserve negative curvature and that singularities mustbe resolved by surgery leaves certain basic questions unanswered. For example,one may ask whether the space of negatively curved metrics on a manifold M isconnected or contractible, or analogous questions for the space of metrics withnegative curvature satisfying various pinching conditions. In higher dimensions,these questions are known to have negative answers [10, 11, 12]; but it has beenconjectured and seems plausible that their answers should be positive if M is ahyperbolic 3-manifold.

D.K. acknowledges NSF support in the form of grants DMS-0505920 and DMS-0545984.

1

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2 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

Such questions have particular relevance to the study of hyperbolic 3-manifolds,because in practice one frequently encounters 3-manifolds on which it is possibleexplicitly to describe a metric of nonconstant negative curvature and to discerncertain geometric information; however, the relationship between this metric andthe hyperbolic metric — which exists by Geometrization — is unclear. Such metricsare constructed by Namazi and Souto in [22], for example, as well as in the famous2π-Theorem of Gromov–Thurston [14], which motivates our work here.

The 2π-Theorem belongs a family of results describing the geometry of “fillings”— that is, manifolds which result from Dehn surgery on cusps of a finite-volumehyperbolic 3-manifold. We describe this construction in greater detail in §2. Theinitial such result, the “hyperbolic Dehn surgery theorem,” [27] asserts that mostfillings of a hyperbolic 3-manifold admit hyperbolic metrics of their own, but it doesnot explicitly describe the set of hyperbolic fillings. The 2π-Theorem remedies thisdeficiency, but with a weaker conclusion, supplying a metric with negative sectionalcurvatures on a filling that satisfies a certain explicit criterion. Another result inthe spirit of the 2π-Theorem is the 6-Theorem due to Agol [2] and, independently,Lackenby [19]. This relaxes the hypotheses of the 2π-Theorem at the expense ofweakening its conclusion to a purely topological statement, one that nonethelessimplies hyperbolizability using Geometrization.

A program more in the spirit of this paper is that of Hodgson–Kerckhoff [15,16, 17], which, under somewhat stronger hypotheses, describes a family of “cone”metrics. These are hyperbolic but singular along an embedded one-manifold, andinterpolate between the original smooth hyperbolic metric and that on the filling.The goal of our program is to give an alternate construction of an interpolatingfamily. We propose using cross curvature flow (xcf) to construct a one-parameterfamily of smooth metrics, with negative but nonconstant sectional curvatures, whichinterpolate between the metric supplied by the 2π-Theorem and the hyperbolicmetric guaranteed by Geometrization.

In [9], Chow and Hamilton introduced the xcf of metrics on a 3-manifold withuniformly signed sectional curvatures. They established certain properties of theflow and conjectured that if the initial datum is a metric with strictly negative sec-tional curvatures, then xcf (suitably normalized) should exist for all time, preservenegative sectional curvature, and converge asymptotically to a metric of constantcurvature. This conjecture would imply contractibility for the space of negativelycurved metrics on a hyperbolic 3-manifold. An additional benefit of a proof of thisconjecture is that it should be easier to track the evolution of relevant geometricquantities in a flow without singularities than it is to track their evolution acrosssurgeries.

xcf is a fully nonlinear, weakly parabolic system of equations, which can bedefined as follows: let Pab = Rab − 1

2Rgab denote the Einstein tensor, and letP ij = giagjbPab. One can define the cross curvature tensor X by

(1.1) Xij =12PuvRiuvj .

Then the cross curvature flow of (M3, g0) is

∂tg = −2X,(1.2)

g(x, 0) = g0(x).(1.3)

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XCF ON A NEGATIVELY CURVED SOLID TORUS 3

Since its introduction in [9], several papers concerning xcf have appeared in theliterature. Buckland established short-time existence of xcf for smooth initialdata on compact manifolds using DeTurck diffeomorphisms to create a strictlyparabolic system [6]. (Like Ricci flow, xcf is only weakly parabolic.) Chen and Mashowed that certain warped product metrics on 2-torus and 2-sphere bundles overthe circle are solutions to xcf [20]. Solutions on locally-homogeneous manifoldshave recently been studied by Cao, Ni, and Saloff-Coste [7, 8], and independentlyby Glickenstein [13]. Two of the authors of this paper have shown that (normalized)xcf is asymptotically stable at hyperbolic metrics [18]. Also, in unpublished earlierwork, Andrews has obtained interesting estimates for more general solutions of xcf.

Here we consider a family of negatively curved metrics on the solid torus D2×S1,with initial data that arise in the proof of the 2π-Theorem. We (1) show short-timeexistence of xcf with prescribed boundary conditions for metrics in this family, (2)establish curvature bounds depending only on the initial data (showing in particularthat negative curvature is preserved for this family), and (3) demonstrate long-timeexistence.

Although they fall short of confirming the Chow–Hamilton conjectures (even formetrics in this family) our results do provide evidence in favor of those conjectures.On one hand, it is encouraging that negative curvature is preserved and long–timeexistence holds. On the other hand, the fully nonlinear character of xcf and thehigh topological complexity of negatively curved manifolds have not yet allowed acomplete proof of convergence to hyperbolic. In fact, we are not yet able to ruleout convergence to a soliton. Note that recent studies of xcf (properly defined) onhomogeneous manifolds admitting curvatures of mixed sign display a much greatervariety of behaviors than the corresponding examples for Ricci flow (cf. [7, 8], [13]).

The purpose of this paper is to introduce our program and to report on partialprogress toward its resolution. The paper is organized as follows. In §2, we givea brief description of the 2π-Theorem and related work. In §3, we describe theproposed program and give some of its motivations. In §4, we derive relevant cur-vature equations and establish appropriate boundary conditions. In §5, we reporton our progress thus far, summarizing the new results established in this paper thatsupport the program. To streamline the exposition, the (sometimes lengthy) proofsof these theorems are collected in §6.

We hope that the results in this paper contribute to further progress in theprogram, and ultimately to further applications of curvature flows to the study ofhyperbolic 3-manifolds.

2. Dehn filling and the 2π-Theorem

Suppose M is a compact 3-manifold with a single boundary component: a torusT 2 ∼= S1 × S1. A closed manifold may be produced from M by Dehn filling ∂M , awell known construction in which a solid torus is glued to M by a homeomorphismof their boundaries. More precisely:

Definition 1. Suppose M is a compact 3-manifold with a torus boundary compo-nent T . Let D2 be the unit disk in R2; let h be a homeomorphism from ∂(D2×S1) =∂D2×S1 to T ; and for p ∈ S1, take λ = h(∂D2×p). Define the manifold M(λ)obtained by Dehn filling M along λ by:

M(λ) =M t (D2 × S1)

x ∼ h(x), x ∈ ∂D2 × S1.

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4 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

We call the isotopy class of λ in T the filling slope, and the image in M(λ) ofD2 × S1 the filling torus.

Clearly, the topology of M(λ) may vary depending on the choice of h, but it isa standard fact that it is determined up to homeomorphism by the choice of fillingslope.

Now suppose that M is a hyperbolic 3-manifold with finite volume. (That is, Mis a smooth manifold equipped with a complete, finite-volume Riemannian metricwhich has sectional curvatures ≡ −1.) It follows from the Margulis Lemma ([21],cf. [26, Theorem 12.5.1]) that if M is noncompact, each of its cusps is diffeomorphicto T 2 × R+. (Here a “cusp” is a component of the complement of a sufficientlylarge compact submanifold-with-boundary.) Thus removing the cusps of M yieldsa compact 3-manifold M whose boundary is a disjoint union of tori.

Suppose for simplicity that M has a single cusp. The hyperbolic Dehn surgeryTheorem, due to Thurston [27, §5], asserts that for all but finitely many choices ofslope λ on ∂M , the closed manifold M(λ) admits a hyperbolic metric. In the pastthirty years, an enormous quantity of work has been devoted to explicitly describingthe set of “hyperbolic filling slopes”, and investigating the relationship between thehyperbolic metrics on M and M(λ). This is the context of the 2π-Theorem, due toGromov and Thurston ([14], cf. [5, Theorem 9]).

A further consequence of the Margulis Lemma is that a parameterization φ : T 2×R+ → M for the cusp of M may be chosen so that the hyperbolic metric on Mpulls back to a warped product metric on T 2 × R+ which restricts on each leveltorus T 2×t to a Euclidean metric. (We will precisely describe this metric below.)We call such a parameterization “good”.

Theorem 1 (Gromov–Thurston). Let M be a one-cusped hyperbolic 3-manifoldwith a good parameterization φ : T 2×R+ →M of the cusp of M , and suppose λ isa slope on ∂M such that φ−1(λ) has a geodesic representative with length greaterthan 2π in some level torus. Then M(λ) admits a Riemannian metric with negativesectional curvatures.

The proof is constructive, yielding a metric on M(λ) which on M is isometric tothe metric inherited fromM , and on the filling torus has variable negative curvature.Below we will sketch a proof, describing the metrics under consideration on D2×S1.It will be convenient to use cylindrical coordinates (µ, λ, r) ∈ [0, 1)× [0, 1)× [0, 1],where with D2 × S1 naturally embedded in R2 × R2, we have

x1 = r cos(2πµ), x2 = cos(2πλ),

y1 = r sin(2πµ), y2 = sin(2πλ).

The core of D2 × S1 is the set (0, 0) × S1 = (0, λ, 0), and a meridian diskis of the form D2 × p = (µ, λ0, r) for fixed p or λ0. The standard rotationsof D2 × S1 are of the form (µ, λ, r) 7→ (µ + µ0, λ, r), fixing the core and rotatingeach meridian disk, or (µ, λ, r) 7→ (µ, λ + λ0, r). All of the metrics we use will berotationally symmetric, and diagonal in cylindrical coordinates, which implies thatthey have the form

(2.1) G(r) = f2(r) dµ2 + g2(r) dλ2 + h2(r) dr2.

Here the functions f , g, and h must satisfy the following regularity conditions atr = 0 to ensure that they extend smoothly across the core.

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XCF ON A NEGATIVELY CURVED SOLID TORUS 5

Lemma 1. G extends to a smooth metric on D2 × S1 if and only if g(r) andh(r) extend to smooth even functions on R with g(0), h(0) > 0, and f extends to asmooth odd function on R with fr(0) = 2πh(0).

Because this is a standard result in Riemannian geometry, we omit the proof.

In a rotationally-symmetric diagonal metric on D2×S1, it will frequently proveconvenient to parameterize by an outward-pointing radial coordinate which mea-sures distance from the core:

(2.2) s(r) =∫ r

0

h(ρ) dρ.

Then G may be written in the form

(2.3) G(s) = f2(s) dµ2 + g2(s) dλ2 + ds2, s ∈ [0, s0].

Here s0 is the distance from the core to the boundary. The smoothness criteriaof the lemma above translate to the requirements that f extends to a smooth oddfunction of s, and g to an even function of s, such that fs(0) = 2π and g(0) > 0.

Example 1. There is a standard description of the cusp of a hyperbolic manifoldas the quotient of a horoball in hyperbolic space H3 by a group of isometries Γ ' Z2.We will use the upper half-space model for H3, namely (x, y, z) ∈ R3 | z > 0 withthe Riemannian metric Gh = z−2(dx2 + dy2 + dz2). Then a standard horoball isH1 = (x, y, z) | z ≥ 1, and the elements of Γ are Euclidean translations fixing thez coordinate, each of the form φ(a,b) : (x, y, z) 7→ (x+ a, y + b, z) for (a, b) ∈ R2.

Without loss of generality, we may assume that Γ is generated by elements of theform φ(M,0) and φ(x0,L), where M,L > 0. Then M is the length of a “meridian”geodesic of the torus T = ∂(H1/Γ) = (x, y, 1)/Γ, and L is the length of an arcjoining the meridian geodesic to itself, perpendicular to it at both ends. (The finalconstant x0 of the definition is a sort of “twisting parameter” and does not enterinto computations using the metrics under consideration here.)

There is a map from the complement of the core in D2 × S1 to H1/Γ, given incylindrical coordinates by

Ψ(µ, λ, r) =(Mµ,Lλ,

1r

), r ∈ (0, 1].

Using Ψ, the hyperbolic metric pulls back to

Ψ∗Gh = M2r2 dµ2 + L2r2 dλ2 +dr2

r2, r ∈ (0, 1].

Using an outward-pointing radial coordinate which measures distance to the bound-ary,

s(r) = −∫ 1

r

dρ = log r,

the hyperbolic metric takes the form

Gcusp = M2e2s dµ2 + L2e2s dλ2 + ds2, s ∈ (−∞, 0].

Sketch of proof of Theorem 1. One seeks a rotationally symmetric diagonal metricon D2 × S1 of the form (2.3), which in addition has negative sectional curvatures.One further requires that in a neighborhood U of ∂(D2 × S1), f and g take theform

(2.4) f(s) = Mes−s0 , g(s) = Les−s0 .

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6 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

If this is the case, then the map h : (µ, λ, s) 7→ (µ, λ, s−s0) takes U isometrically toa neighborhood of ∂(H1/Γ) with the metric Gcusp. Note also that for any p ∈ S1,λ = h(∂D2 × p) is a geodesic on T with length M ; thus in this case, there is ametric on M(λ) yielding the theorem. A computation establishes that the sectionalcurvatures of G(s) are as follows:

Kλµ = −fsgsfg

, Kλs = −gssg, Kµs = −fss

f.

Since f and g are positive on (0, s0], and fs(0) = 2π, it follows that all sectionalcurvatures are negative if and only if fs, gs, fss, and gss are positive on (0, s0]. Sincethe components of G must satisfy (2.4) near the boundary, one has fs(s0) = M .Thus since fss > 0, M must be greater than fs(0) = 2π. One can show that thisnecessary condition is also sufficient for there to exist a metric of the form (2.3)satisfying our conditions. (See [5, Theorem 9].)

3. Description of the program

We now outline our proposed program. As was discussed above, the 2π-Theoremconstructs a negatively curved metric on certain 3-manifolds obtained by Dehnfilling. By Geometrization, any such manifold admits a hyperbolic metric. Ourgoal is to use xcf to construct a smooth one-parameter family of metrics betweenthe “2π-metric” and the hyperbolic metric. Motivated by the Hodgson–Kerckhoffconstructions [15, 16, 17], one might expect the program to work for slopes λ on∂M such that φ−1(λ) has a geodesic representative of length ≥ L, for some L > 2π.(Because our method depends on a stability result, one does not expect to be ableto take L = 2π.)

The first step is to take as an initial datum a negatively curved metric G(s, 0)on a solid torus given by equation (2.3). We evolve this metric by cross curvatureflow. Because this is a manifold-with-boundary, we impose time dependent Dirichletboundary conditions corresponding to a homothetically evolving hyperbolic metric.(Example 2 below motivates this choice.) In this paper, we show that the flow existsfor all time and preserves negative sectional curvatures. Inspired by the conjectureof Hamilton and Chow, we conjecture that (after normalization) G(·, t) convergesuniformly to a hyperbolic metric in C2,α as t → ∞, at least on sets compactlycontained in the complement of the boundary, i.e. on [0, 1)× [0, 1)× [0, r∗] for anyr∗ ∈ (0, 1). The results in this paper support this conjecture.

Once the conjecture is proved, the second step of the program is to homoth-etically evolve the negatively curved metric on M(λ) that is guaranteed by the2π-Theorem. Recall that the metric on M is isometric to the hyperbolic metricinherited from M . The Dirichlet boundary conditions on the solid torus are de-liberately chosen to match the boundary conditions on M . Once one has shownthat the gluing map is C2+α, one may apply the asymptotic stability of hyperbolicmetrics under cross curvature flow (proved by two of the authors [18]) to establishexponential convergence to a metric of constant negative curvature.

A successful completion of this program would exhibit a one-parameter familyof negatively curved metrics connecting the “2π metric” on M(λ) to the hyperbolicmetric that, according to Geometrization, this manifold must admit.

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XCF ON A NEGATIVELY CURVED SOLID TORUS 7

4. Basic equations and boundary conditions

Let M3 = D2×S1 be the solid torus, a 3-manifold with boundary. As above, werestrict our attention to evolving metrics G on M3 which are rotationally symmetric.These are metrics that are diagonal in cylindrical coordinates, and for which theisometry group acts transitively on each torus consisting of the locus of points afixed distance from the core. We may write G as

G(s, t) = f2(s, t) dµ2 + g2(s, t) dλ2 + ds2(4.1a)

= e2u(s,t) dµ2 + e2v(s,t) dλ2 + ds2,(4.1b)

where s is arclength measured outward from the core, as defined in (2.2).The following example describes the prototypical negatively curved diagonal met-

ric on a solid torus: the hyperbolic metric of constant curvature −1. This explainsour choice of boundary conditions.

Example 2. The metric on a regular neighborhood of a closed geodesic in a hyper-bolic manifold is given by

Ghyp =(

21− r2

)2(r2 dµ2 +

b

4(1 + r2)2 dλ2 + dr2

)for r ∈ (0, 1), where b is the length of the geodesic. Using the geodesic radialcoordinate

s(r) =∫ r

0

21− ρ2

dρ = log(

1 + r

1− r

),

this takes the form

Ghyp = 4π sinh2 s dµ2 + b cosh2 s dλ2 + ds2.

The solution of xcf on any closed hyperbolic manifold with initial data G0 isG(t) =

√1 + 4tG0. On the solid torus, the solution of xcf with initial data Ghyp

is similarly G(t) =√

1 + 4tGhyp.

The principal sectional curvatures of the metric G given by (4.1) are

Kλµ = −fsgsfg

= −usvs,(4.2a)

Kλr = −gssg

= −(vss + v2s),(4.2b)

Kµr = −fssf

= −(uss + u2s).(4.2c)

We are interested in the case that G has negative sectional curvatures. Thisoccurs when f and g, in addition to being positive, are monotonically increasingconvex functions of s that satisfy the following conditions at the origin:

lims→0

gsf

= lims→0

gssfs

= lims→0

gss > 0,(4.3a)

lims→0

fssf

= lims→0

fsssfs

= lims→0

fsss > 0.(4.3b)

One finds from the first equality in (4.3) that the sectional curvatures Kµλ andKrλ are identical at the core. This reflects the fact that rotation about the core isan isometry, and so the sectional curvatures tangent to any two planes containing∂∂λ are equal.

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8 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

Henceforth, we assume the initial metric G0 has negative sectional curvatures.We denote the absolute values of the principal sectional curvatures by α, β, and γ,namely

(4.4) α = −Kλµ, β = −Kλr, γ = −Kµr.

The assumption of negative sectional curvature imposes certain constraints onthe initial values of f , g and h, as mentioned above. Namely, we require thatf(1, 0) = `1 > 2π and that the initial radius r0 = s(1, 0) satisfies 1 < r0 < `1/2π.We also assume that the metricG0 = G(·, 0) obeys a global C2+θ bound and thatG0

has constant negative sectional curvature at the core — namely, that α = β = γ > 0at r = 0.

We now apply cross curvature flow to a metric of the form (4.1). The evolutionof this metric under xcf is equivalent to the system

ft =fsgsfg

fss,(4.5a)

gt =fsgsfg

gss,(4.5b)

ht =fssgssfg

h.(4.5c)

Notice that these equations remain nondegenerate only as long as fsgs/(gf) remainsstrictly positive.

Sometimes, it is more tractable to work with the equations for

(4.6) u = log f, v = log g, w = log h.

These quantities evolve by

ut = αγ = usvsuss + u3svs(4.7a)

vt = αβ = usvsvss + v3sus(4.7b)

wt = βγ = ussvss + v2suss + u2

svss + u2sv

2s ,(4.7c)

Remark 1. Given w(·, 0), the evolution of w is determined by us, vs, uss, and vss.So we may (and usually do) suppress its equation in what follows.

Remark 2. Observe that system (4.5) for f and g — equivalently, system (4.7)for u and v — is strictly parabolic. This is because our choice to parameterize byarclength s fixes a gauge and thereby breaks the diffeomorphism invariance of xcf.This effectively replaces the DeTurck diffeomorphisms in our proof of short-timeexistence (Theorem 2, below). This simplification is possible because, in contrastto the more general situation considered by Buckland [6], the spatial dependence ofour metrics is only on the radial coordinate r ∈ [0, 1]. A similar situation occursfor rotationally invariant Ricci flow solutions; compare the system for ϕ and ψconsidered in [4], for example.

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XCF ON A NEGATIVELY CURVED SOLID TORUS 9

Because the solid torus is a manifold-with-boundary, one must prescribe bound-ary conditions. We prescribe the Dirichlet boundary conditions given by

u(x, t) = log(f(1, 0)(1 + 4t)1/4

)(4.8a)

v(x, t) = log(g(1, 0)(1 + 4t)1/4

)(4.8b)

w(x, t) = log(

(1 + 4t)1/4).(4.8c)

As noted above, our choice of boundary conditions are those attained by a homo-thetically evolving hyperbolic metric.

5. Summary of results

We now state our results that establish long-time existence and preservation ofnegative sectional curvature for the metrics under consideration in this paper. Tostreamline the exposition, we postpone the proofs until the next section.

5.1. Long-time existence and uniqueness. We first establish short-time exis-tence and uniqueness. Since we are considering xcf on a manifold-with-boundary,these results do not immediately follow from [6]. So we use the machinery of [1].Fix θ ∈ (0, 1).

Definition 2. Given τ > 0, let Eτ denote the space

Eτ := C1+θ([0, τ ], C0(Ω, CN )

)∩ L∞

([0, τ), C2+2θ(Ω, CN )

),

endowed with the norm

|| ~X||Eτ := supt∈[0,τ ]

|| ~X||C0 + [Dt~X]Cθ + sup

t∈[0,τ ]

[D2x~X]C2θ .

Here, Cr+θ denotes the usual Holder norm, and L∞ has its standard meaning.

Theorem 2. There exists τ0 > 0 such that the fully nonlinear parabolic system(4.7) has a unique solution (u, v, w) ∈ Eτ0 .

It is not known in general whether solutions to cross curvature flow exist forall time. However, in the particular case of our rotationally symmetric metric onthe solid torus, we can show a priori bounds on the derivatives of the sectionalcurvatures.

Theorem 3. Let G be a rotationally symmetric metric on the solid torus given byequation (4.1). For every τ > 0 and m ∈ N, there exists a constant Cm dependingon m, τ, and K such that if

β(s, t), γ(s, t) ≤ K for all s ∈M and t ∈ [0, τ ],

then ∣∣∣∣ ∂m∂sm β(s, t)∣∣∣∣ ≤ Cm

tmfor all s ∈M and t ∈ [0, τ ],

and ∣∣∣∣ ∂m∂sm γ(s, t)∣∣∣∣ ≤ Cm

tmfor all s ∈M and t ∈ [0, τ ].

We use these estimates together with the fact that the curvatures stay boundedto show, in the usual way, long-time existence of solutions to equations (4.5).

Theorem 4. The solution (u, v, w) to xcf on the solid torus exists for all time.

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10 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

5.2. Curvature estimates. One can compute the following evolution equationsfor the sectional curvatures:

αt = ααss + [2α(us + vs) + βus + γvs]αs + 2α(α2 − 2βγ)(5.9a)

βt = αβss + (3βus + γvs − 2αus)βs + 2β[u2s(α− β)− αγ](5.9b)

γt = αγss + (3γvs + βus − 2αvs)γs + 2γ[v2s(α− γ)− αβ].(5.9c)

This system is well behaved as long as α > 0, since this makes α ∂2

∂s2 an ellipticoperator. In particular, a maximum principle then applies at interior points. Wesay s(r, t) is an interior point if r ∈ (0, 1).

Our goal, in the context of §3, is to use these evolution equations to show con-vergence to hyperbolic. Here we collect various estimates that represent progresstoward that goal. Define

K0 = supM×0

α, β, γ and L0 = infM×0

α, β, γ.

Theorem 5.(1) For as long as a solution exists, α > 0. Thus the xcf operator remains

elliptic.(2) The quantities α, β, and γ are bounded from above by K0.(3) The quantities α, β, and γ are bounded from below by L0e

−4K20 t.

(4) Negative sectional curvature is preserved.(5) Over the course of the xcf evolution, one has

α ≥ L0

4K0L0t+ 1.

Finally, we follow [9] to obtain evidence that our solution converges to a hyper-bolic metric in an integral sense. We use their notation and define

(5.10) J =∫M

(P

3− (detP )

13 ) dV ,

where P = gijPij . Notice the integrand is nonnegative (by the arithmetic-geometric

mean inequality) and is identically zero if and only if gij has constant curvature.Hamilton and Chow’s theorem does not directly apply to our setting, as we have amanifold with boundary. However, we are able to prove the analogous theorem.

Theorem 6. Under xcf of a rotationally-symmetric metric on the solid torus, onehas

(5.11)dJ

dt≤ 0.

6. Collected proofs

6.1. Proofs of long-time existence and uniqueness. We first use the theory of[1] for short-time existence and regularity of solutions to fully nonlinear parabolicsystems. Consider the second-order system

∂t ~X = F (t, x, ~X,D ~X,D2 ~X) for (t, x) in [0, T ]× Ω

H(t, x, ~X,D ~X) = 0 for (t, x) in [0, T ]× ∂Ω

~X(0, x) = φ(x) for x in Ω

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XCF ON A NEGATIVELY CURVED SOLID TORUS 11

where T > 0, Ω is a bounded, smooth domain of Rn, and F,H are smooth CN -valued functions.

The short-time existence theory requires the following hypotheses:Regularity: The boundary of Ω and the functions F,H, φ satisfy

∂Ω ∈ C2+2θ, φ ∈ C2+2θ(Ω,CN )

F ∈ C2(Γ,CN ),

H ∈ C3(Γ′N ), (θ ∈ (0, 1/2)),

where Γ := [0,∞)× Ω× CN × CnN × Cn2N and Γ′ := [0,∞)× Ω× CN × CnN .Compatibility: The initial datum φ satisfies

H(0, x, φ(x), Dφ(x)) = 0, x ∈ ∂Ω.

If these hypotheses are satisfied, one has:

Theorem 7. There exists τ0 > 0 such that the fully nonlinear parabolic system hasa unique solution ~X ∈ Eτ0 , where

Eτ := C1+θ([0, τ ], C0(Ω, CN )

)∩ L∞

([0, τ), C2+2θ(Ω, CN )

),

with the norm

|| ~X||Eτ := supt∈[0,τ ]

|| ~X||C0 + [Dt~X]Cθ + sup

t∈[0,τ ]

[D2x~X]C2θ .

Proof of Theorem 2. As explained in Remark 2, the parameterization by arclengths, as defined in (2.2), makes xcf a strictly parabolic system; furthermore, thisparameterization respects the stated Dirichlet boundary conditions.

Thus we may apply Theorem 7 to the system of equations (4.7) with boundaryconditions given by (4.8). Clearly, the regularity hypotheses for F and H aresatisfied, because our initial data are C∞. Moreover, one has u(x, 0) = log f(1, 0),v(x, 0) = log g(1, 0), and w(x, 0) = 0 for x on the boundary torus, so that thecompatibility hypothesis is likewise met. Hence Theorem 2 follows.

The proof of long-time existence of solutions to the flow is standard once weobtain a priori derivative estimates. So we will first outline the proof of Theorem 3.We prove this using estimates of Bernstein–Bando–Shi type.

Sketch of proof of Theorem 3. Define M+ := tβs + λ1β2 + λ2γ

2, where λ1 and λ2

are two constants to be chosen later. (We will see that they depend only on theinitial conditions and the length of the time interval under consideration.) Whenone computes the evolution of M+, one sees that it has the following structure:

∂M+

∂t= α(M+)ss + (M+)sF1 + 3t(βs)2us + tβsγsvs

+ tβsF2 + tγsF3 − 2αλ1(βs)2 − 2αλ2(γs)2,

where Fi = Fi(α, β, γ) are polynomials depending on the curvatures.Now let τ > 0 be some real number, and consider the evolution equation on [0, τ ].

Using bounds on the curvatures and multiple applications of Cauchy–Schwarz, onecan show that

∂M+

∂t≤ α(M+)ss + (M+)sF1 + (βs)2(F2 − 2Kλ1) + (γs)2(F3 − 2Kλ2) + F4,

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12 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

where K > 0 is the lower bound for α on [0, τ ], and the Fi (which may differ fromabove) also depend on τ .

Since all of the Fi are bounded from above, we choose λ1 and λ2 so that theterms F2 − 2Kλ1 and F3 − 2Kλ2 are both negative. Then we have

(M+)t ≤ α(M+)ss + (M+)sF1 + C,

where C is a constant depending only on the initial data. Thus by the parabolicmaximum principle, we have

supx∈M3

M+(x, t) ≤ Ct+D ≤ Cτ +D,

for all t ∈ [0, τ ], where again D is a constant just depending on the initial data.Hence, on this bounded time interval,

βs ≤C

t.

Notice that in an analogous fashion, we can obtain a lower bound for βs on abounded time interval by considering M− := −tβs − λ1β

2 − λ2γ2. Thus we obtain

our desired result that |βs| ≤ Ct on [0, τ ], where C depends only on the initial data

and on τ .Similarly, to estimate γs, one considers N+ := tγs+λ1β

2 +γ2 along with N− :=−tγs − λ1β

2 − λ2γ2 and shows that in fact |γs| ≤ C

t , for C as above.We then use induction to show that higher-order estimates |∂

mβ∂sm |, |

∂mγ∂sm | ≤

Ctm

hold on [0, τ ] for any m > 0.

In order to show long-time existence of solutions to the system given by (4.5),we will prove a theorem analogous to that for Ricci flow.

Theorem 8. Let G0 be a metric on the solid torus M3 given by (4.1). Thenunnormalized cross curvature flow has a unique solution G(t) such that G(0) = G0.This solution exists on a maximal time interval [0, T ). If T ≤ ∞, then at least oneof

limt→T

( sups∈M3

|α(s, t)|), limt→T

( sups∈M3

|β(s, t)|), limt→T

( sups∈M3

|γ(s, t)|)

is infinite.

Proof. As for Ricci flow, we prove the contrapositive of this statement. Supposethat the solution exists on a maximal time interval [0, T ), and that there existsK > 0 such that sup0≤t<T |α(s, t)|, sup0≤t<T |β(s, t)|, and sup0≤t<T |γ(s, t)| ≤ K.The key idea is to show that G(s, T ) is a smooth limit metric on the solid torus ofthe form given in (4.1). We can define f(T ) and g(T ) to be

f(s, T ) = f(s, τ) +∫ T

τ

αγf(s, t) dt

g(s, T ) = g(s, τ) +∫ T

τ

αβg(s, t) dt ,

where τ ∈ [0, T ) is arbitrary. Using this formulation to compute derivatives off(T ) and g(T ), one can use Theorem 3 to bound the curvature quantities, therebyshowing that both f(T ) and g(T ) are smooth. It remains to show that the metricG(T ) = f(T )2 dµ2 + g(T )2 dλ2 +ds2 extends to a smooth metric on the solid torus;namely, that g(T ) extends to an even function such that g(0, T ) > 0 and thatf(T ) extends to an odd function with fs(0, T ) = 1. These facts can be seen from

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XCF ON A NEGATIVELY CURVED SOLID TORUS 13

the integral formulation above, if one recalls that the curvatures extend to evenfunctions. Then G(T ) is a smooth metric on the solid torus, so Theorem 2 impliesthat a solution exists on [T, T + ε) for some ε > 0. This contradicts T beingmaximal.

Corollary 1 (Theorem 4). Let G0 be as in (4.1). Then the solution G(t) to crosscurvature flow with G(0) = G0 exists for all time.

Proof. By Parts (2) and (3) of Theorem 5, we obtain uniform bounds for α, β, γ.Thus Theorem 8 implies T =∞, to wit, that the flow exists for all time.

6.2. Analysis of core conditions. Recall that one of our requirements on themetric G was that f = 0 on the core circle. Then the form in which the evolutionequations (4.7) are written involves the quantity us = fs/f , which blows up ass 0. Because of this, we treat the core as a special case for the curvatureevolution equations. We do this in a standard way using l’Hopital’s rule, beginningwith the following lemma.

Lemma 2. On the solid torus, one has

lims→0

u2s(α− β) = lim

s→0

13

(βγ − β2 − βss).

Proof. Writing the expression us(α− β) in terms of f and g and their derivatives,we obtain

u2s(α− β) =

f2s

f2

(fsgsfg− gss

g

)=f2s

g

(fsgs − fgss

f3

).

The quantity outside the parentheses above has a well-defined limit as s → 0,and both the numerator and denominator of the fraction inside the parenthesesapproach 0 as s→ 0. We apply l’Hopital’s rule three times to obtain the following:

lims→0

u2s(α− β) = lim

s→0

fsg

(fssgs − fgsss

3f2

)= lims→0

1g

(fsssgs + fssgss − fsgsss − fgssss

6f

)= lims→0

13

(fsss

gssg− gssss

g

).

A quick computation of the limit of βss as s→ 0 reveals that

lims→0

βss = lims→0

gssssg− g2

ss

g2,

and this together with the fact that fsss and γ have the same limit as s→ 0 yieldsthe conclusion.

From the result above, the following description of the curvature evolution atthe core follows readily.

Lemma 3. At the core, the evolution equations for the curvatures are as follows:

αt = 4ααss + 2α3 − 4α2γ(6.1)

βt =43αβss −

23β3 − 4

3β2γ(6.2)

γt = 2αγss − 2α2γ.(6.3)

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14 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

Proof. We recall that the curvatures all extend past the origin to smooth evenfunctions of s. Using this fact and l’Hopital’s rule, we have

lims→0

usαs = lims→0

αss,

with similar identities holding for β and γ. The lemma follows readily from thisfact and Lemma 2, using the fact that α = β at the core.

Remark 3. It may be noted that since α = β at the core, their evolution equationsshould be identical. In fact this is true; the missing ingredient is the fact that

lims→0

αss = lims→0

13

(βss + 2βγ − 2β2).

This may be proved by explicitly writing the formula for αss in terms of f and gand their derivatives, and using l’Hopital’s rule to simplify the limit as s → 0, asabove.

6.3. Proofs of curvature estimates. Here we provide the proofs of the curvatureestimates, using the maximum principle.

Lemma 4. As long as a smooth xcf solution exists with α > 0, one has β ≥ 0and γ ≥ 0.

Proof. This is a standard maximum principle argument, slightly tweaked to accom-modate the core and boundary conditions. We give the proof for β; the proof forγ is similar. Suppose a smooth solution with α > 0 exists on the time interval[0, τ ]. Let C0 be the maximum attained by the function u2

s(α − β) − αγ on thistime interval. (Recall that although us blows up at the origin, u2

s(α − β) is atleast continuous there by Lemma 2.) Define C1 to be the maximum achieved by−2/3(β2 + 2βγ) at the core on the time interval [0, τ ], and let C be the maximumof C0 and C1. Define φ(s, t) = e−Ctβ(s, t). The evolution equation for φ is

φt = αφss + (3βus + γvs − 2αus)φs + φ(u2s(α− β)− αγ − C)

on the interior, and at the core it is

φt =43αφss + φ(−2

3(β2 + 2βγ)− C).

At a local minimum for φ in the interior or at the core, one has φss ≥ 0 and φs = 0;and from our definition of C, it follows that if φ < 0 at such a local minimum, thenφt ≥ 0 there.

For δ > 0, consider the function φ+ δ(t+1). Initially this function has all valueslarger than 0, since β has all values larger than 0 initially. If there is a first timein (0, τ ] that φ(s, t) + δ(t + 1) = 0 for some s, then the observations above implythat (φ + δ(t + 1))t = φt + δ ≥ δ there. (Such a minimum cannot occur on theboundary, since values of β are always positive there.) But since t is the first timethat such a minimum occurs, computing the time derivative (φ + δ(t + 1))t frombelow shows that this quantity must be less than or equal to 0, a contradiction.Thus φ+ δ(t+ 1) is positive on [0, τ ], and since δ > 0 is arbitrary, so is φ. But thenso is β, since φ is a positive multiple of β.

Lemma 5. Suppose a solution exists for 0 ≤ t ≤ T . Define K = supM3×[0,T ]β, γ.Then for all t ∈ [0, T ], one has

αmin(t) ≥ αmin(0)e−4K2t.

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XCF ON A NEGATIVELY CURVED SOLID TORUS 15

Proof. Note that αmin(0) ≤ K, since the curvatures are all equal to −1 at the originat time t = 0. For δ > 0, define the barrier function A(t) = αmin(0)e−4K2t − δ.Then α ≥ A+ δ at t = 0. If there is a first time t ∈ (0, T ] such that αmin(t) = A(t),then αmin(t) is attained either at an interior point or the core.

In the former case, one has αt ≤ A′,ααss = Aαss ≥ 0,αs = 0.

By hypothesis β ≤ K and γ ≤ K. Hence at (s, t), one has

A′2(A+ δ) ≥ αt > −4αβγ ≥ −4K2A.

This is evidently impossible.If the minimum occurs at the core, then appealing to the evolution equation

there shows that αt ≥ −4α2γ ≥ −2A2K. This yields

A′2(A+ δ) ≥ αt ≥ −2A2K.

Since K ≥ αmin(0) and K ≥ 1 we have K > A at t. Plugging into the aboveinequality yields

A′2(A+ δ) ≥ −4A2K > −4K2A.

This is plainly a contradiction. Since δ was arbitrary and α is controlled on theboundary, the lemma follows.

Corollary 2 (Part (1) of Theorem 5). For as long as a solution exists, α > 0.

This follows immediately from the lemma, and shows that the xcf operatorremains elliptic for as long as the flow exists.

Now let K0 = supM×0α, β, γ.

Lemma 6. Suppose a solution exists for 0 ≤ t ≤ T . Define

K ′ = max K0, supM×[0,T ]

α.

Then with K as in Lemma 5, we have K ≤ K ′.

Proof. We first consider the case of γ. The key observation is that at an interiormaximum (s, t) with γ(s, t) ≥ α, we have

γt(s, t) ≤ 2β(v2s(α− γ)− αβ) ≤ −2αβγ ≤ 0.

This follows from the evolution equation for γ after observing that at such a point,one has αγss ≤ 0 and γs = 0. The corresponding fact at the core follows from theevolution equations in Lemma 3.

Now for δ > 0, define γδ(s, t) = γ(s, t) − δ(t + 1). Then γδ is initially smallerthan K ′. If there is a first time t with γδ(s, t) = K ′, then by the above we have(γδ)t ≤ −δ at such a point. But we must have (γδ)t ≥ 0, a contradiction. Sinceδ > 0 was arbitrary and γ is controlled on the boundary, the lemma follows.

The proof for β is analogous.

Notice that the estimates of Lemmas 5 and 6 do not give a priori bounds (interms of the initial data) on the curvatures, as they ultimately rely on the upperbound attained by α over the course of the evolution. We now show that an upperbound is in fact the quantity K0.

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16 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

Lemma 7. For as long as the flow exists, α(s, t) ≤ K0.

Proof. Define αδ(s, t) = α(s, t)− δ(t+ 1). Then αδ(·, 0) < K0. Suppose there is afirst time t0 at which αδ attains the value K0. We claim that the value K0+δ(t0+1)attained by α at δ0 is the maximum value it attains on the interval [0, t0]. To seethis, suppose there were some t < t0 such that α(s, t) = K0 + δ(t0 + 1) for somes. Then αδ(s, t) = K0 + δ(t0 − t) > K0, a contradiction to our hypothesis thatt0 is the first time αδ attains the value K0. Thus by Lemma 6, the values of βand γ are bounded above by K0 + δ(t0 + 1) on the interval [0, t0]. In particular(this is the important point), at the maximum of α at time t0, we have α ≥ βand α ≥ γ. Appealing to the evolution equation for α recorded in (5.9), one mayobtain a contradiction in the usual way if the quantity α2 − 2βγ < 0. The difficultcase (and the reason we have not given a priori bounds to this point) is when thisinequality does not hold. To deal with this, we use the fact that α = usvs to rewritethe evolution equation for α. Recall that αs = βus + γvs − α(us + vs). Using this,we rewrite the evolution equation for α as follows:(6.4)αt = ααss + [βus + γvs]αs + 2α[(us + vs)αs + α2 − 2βγ]

= ααss + [βus + γvs]αs + 2α[u2s(β − α) + v2

s(γ − α) + αβ + αγ − α2 − 2βγ]= ααss + [βus + γvs]αs − 2α[u2

s(α− β) + v2s(α− γ) + (α− β)(α− γ) + βγ].

At the maximum for α at time t0, we have already observed that β ≤ α and γ ≤ α.Therefore the zero-order term of the differential equation (6.4) is nonpositive, and acontradiction is obtained using the maximum principle in the standard way. Sinceδ > 0 was arbitrary, the Lemma is proved if the maximum occurs in the interior. Ifthe maximum occurs at the core, we note since α = β there and β ≤ K0 +δ(t0 +1),this must also be a local maximum for β. Using the evolution equation for β at thecore, we find that βt = αt ≤ 0, also contradiction. Hence the lemma is proved.

The following theorem is an immediate corollary of Lemma 6 and Lemma 7.

Theorem 9 (Part (2) of Theorem 5). For as long as the flow exists, α, β, and γare bounded above by K0.

The universal upper bound of Theorem 9 may be used to give a universal lowerbound for the sectional curvatures, as in Lemma 5.

Theorem 10 (Part (3) of Theorem 5). Let L0 = infM×0α, β, γ . For as longas the flow exists, α, β, and γ are bounded below by L0e

−4K20 t.

Proof. The case of α follows immediately from Lemma 5, after noting that Theo-rem 9 implies that the constant K in the Lemma is less than or equal to K0, andthat αmin(0) ≥ L0.

We next address the case of γ. As in the proof of Lemma 5, for δ > 0 we definea barrier function A(t) = L0e

−4K20 t − δ. Then γ > A at t = 0. If there is a first

time t > 0 such that γmin(t) = A(t), then at such a point one has γt ≤ A′,αγss ≥ 0,γs = 0.

Note that at this point, one has α > γ, since the inequality holds for α. If theminimum occurs in the interior, then by the evolution equation satisfied by γ there

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XCF ON A NEGATIVELY CURVED SOLID TORUS 17

and the fact that α ≤ K0 and β ≤ K0, we have

A′ = −4K20 (A+ δ) ≥ γt > −2αβγ ≥ −2K2

0A.

This is a contradiction. If the minimum occurs at the core, appealing to the evolu-tion equation there (see Lemma 3) yields

A′ = −4K20 (A+ δ) ≥ γt > −2α2γ ≥ −2K2

0A,

again yielding a contradiction. Since δ > 0 is arbitrary the result is proved for γ.The case of β is analogous.

Theorems 9 and 10 combine to give bounds for the sectional curvatures that holdfor all positive times.

Theorem 10 has another important consequence:

Corollary 3 (Part (4) of Theorem 5). For as long as a the solution of xcf exists,negative sectional curvature is preserved.

We can slightly improve the lower bound for α.

Lemma 8. The evolution equation for α in the interior may be written in thefollowing forms:

αt = ααss + [βus + γvs + 2α(us + vs)]αs + 2α(α2 − 2βγ)(6.5a)

αt = ααss + [γvs − 3βus + 2α(us + vs)]αs + 4u2sβ(β − α)− 2α2(2β − α)

(6.5b)

αt = ααss + [βus − 3γvs + 2α(us + vs)]αs + 4v2sγ(γ − α)− 2α2(2γ − α)

(6.5c)

αt = ααss + [βus + γvs]αs − 2α[u2s(α− β) + v2

s(α− γ) + (α− β)(α− γ) + βγ].(6.5d)

Proof. Equations (a) and (d) above were previously obtained; they are equations(5.9a) and (6.4), respectively. Equation (b) is obtained from (a) by separating offa factor of 4βusαs, so that one obtains

αt = ααss + [γvs − 3βus + 2α(us + vs)]αs + 4βusαs + 2α3 − 4αβγ.

It is easily computed that βusαs = βu2s(β − α) + αβγ − α2β. Substituting this

in the equation above and simplifying yields the result. Equation (c) is obtainedanalogously, but by separating off a factor of 4γvsαs instead of 4βusαs.

We use the new equations below to improve the lower bound on the decay of αfrom exponential to polynomial. Recall that K0 is the supremum of the curvaturesof the initial metric, and that L0 is the infimum.

Theorem 11 (Part (5) of Theorem 5). Over the course of the evolution, one has

α ≥ L0

4K0L0t+ 1.

Proof. For δ > 0, we define the barrier function

A(t) =L0

4K0L0t+ 1− δ,

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18 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

and note that α(·, 0) > A(0). Suppose there is a first time t > 0 at which α(s, t) =A(t) for some s. If this point s is in the interior, the fact that it is a minimum forα implies that αss ≥ 0 and αs = 0 there. Furthermore, we must have

αt ≤ A′(t) = −4K0(A+ δ)2.

At this point the analysis breaks up into three cases depending on the relationshipof α with the other curvatures.

Suppose first that α(s, t) < β(s, t). Then using equation (6.5a), we see from theabove that

−4K0(A+ δ)2 ≥ αt = 4u2sβ(β − α)− 2α2(2β − α) > −4K0α

2 = −4K0A2.

This is a contradiction.If α(s, t) < γ(s, t), an analogous analysis using equation (6.5b) yields a contra-

diction in the same way.It remains to consider the case that α ≥ β, γ. In this case, we use the original

evolution equation (5.9) for α. This yields

−4K0(A+ δ)2 ≥ αt = 2α(α2 − 2βγ) ≥ −2α3 ≥ −2K0A2,

and again a contradiction is obtained.Finally, if the minimum occurs at the core, appealing to the evolution equation

there (see Lemma 3) yields

αt = −2α2(2γ − α) ≥ −4K0α2,

and an analysis similar to the above yields a contradiction. Since δ > 0 wasarbitrary, the result is proved.

6.4. Integral convergence to hyperbolic. Here we prove Theorem 6. Recallthat we defined

J =∫M

(P

3− (detP )

13 ) dV

in (5.10), where P here denotes the trace of the Einstein tensor. Notice thatthe integrand is nonnegative (by the arithmetic-geometric mean inequality) and isidentically zero if and only if the metric has constant curvature.

The only difficulty is that we are considering a manifold-with-boundary. Becausethe proof of monotonicity in [9] relies on integration by parts, we must be able tocontrol the boundary terms that arise in our situation. In what follows, we adopttheir notation and use the following result.

Lemma 9 (Chow–Hamilton). The evolution of the Einstein tensor under xcf is

(6.6)∂

∂tP ij = ∇k∇l(P klP ij − P ikP jl)− detPgij −XP ij ,

where X = gijXij is the trace of the cross curvature tensor.

Proof. See [9].

Lemma 10. For every smooth φ defined on our solid torus solution, one has

(6.7)d

dt

∫ si(t)

0

φds =∫ s1(t)

0

(φt + βγ) ds .

Proof. Straightforward computation.

We now prove the final result of this paper.

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XCF ON A NEGATIVELY CURVED SOLID TORUS 19

Proof of Theorem 6. We begin by computing

d

dt

∫ s1(t)

0

P ds =∫ s1(t)

0

(Pt + βγP ) ds

=∫ s1(t)

0

gij(∇k∇l(P klP ij − P ikP jl))

−3 detP −XP + 2XijPij + βγP

ds

=∫ s1(t)

0

(3 detP − αβP − αγP ) ds + 〈 ∂∂s, P kl∇lP − P jl∇lP kj 〉

∣∣∣∣s1(t)0

.

In terms of our metric, the boundary term becomes[α(βs + γs)−

βfs(α− β)f

− γgs(α− γ)g

]∣∣∣∣s1(t)0

.

Let Vij denote the inverse of P ij . Then

d

dt

∫ s1(t)

0

(detP )13 ds =

∫ s1(t)

0

(∂t(detP )13 + βγ(detP )

13 ) ds

=∫ s1(t)

0

(detP )13 (

13Vij∇k∇l(P klP ij − P ikP jl)) ds

+∫ s1(t)

0

(detP )13 (

13Vij(−detPgij −XP ij) + 2X) + βγ) ds

=: I1 + I2.

Let us first consider I2. Because V = (detP )−1X, one has∫ s1(t)

0

(detP )13 (

13Vij(− detPgij −XP ij) + 2X) + βγ) ds

=∫ s1(t)

0

(detP )13 (

13X − αγ − αβ) ds .

Now we do integration by parts on I1 to obtain

I1 = −12

∫ s1(t)

0

∇k((detP )1/3Vij)(P kl∇lP ij − P jl∇lP ik) ds

+⟨

(detP )1/3Vij ⊗∂

∂s, P kl∇lP ij − P jl∇lP ik

⟩∣∣∣∣s1(t)0

=13

∫ s1(t)

0

(12

∣∣Eijk − Ejik∣∣2V

+16|T i|2V

)(detP )1/3 ds

+⟨

(detP )1/3Vij ⊗∂

∂s, P kl∇lP ij − P jl∇lP ik

⟩∣∣∣∣s1(t)0

,

where Eijk, et cetera, have the same meanings as in [9].A computation shows that the boundary term is

〈(detP )13Vij ⊗

∂s, P kl∇lP ij − P jl∇lP ik〉

∣∣∣∣s1(t)0

= (detP )13VijT

1ij∣∣∣s1(t)0

= (αβγ)13 (α

ββs +

α

γγs +

fsf

(β − α) +gsg

(γ − α)∣∣∣∣s1(t)0

.

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20 JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

Now we collect all of the terms above to see that

d

dtJ =

∫ s1(t)

0

(detP − 13αβP − 1

3αγP − (detP )

13 (X

3− αγ − αβ) ds

= −13

∫ s1(t)

0

(12|Eijk − Ejik|2V +

16|T i|2V )(detP )

13 ds

+ α(βs + γs)−βfs(α− β)

f− γgs(α− γ)

g

∣∣∣∣s1(t)0

− (αβγ)13 (α

ββs +

α

γγs +

fsf

(β − α) +gsg

(γ − α)∣∣∣∣s1(t)0

.

Hence

d

dtJ ≤ −

∫ s1(t)

0

(detP )13 (X

3− (deth)

13 ) ds

−∫ s1(t)

0

αβ(P

3− (detP )

13 ) ds

−∫ s1(t)

0

αγ(P

3− (detP )

13 ) ds

+

α(βs + γs)− βfs(α−β)

f − γgs(α−γ)g

−(αβγ)13 (αβ βs + α

γ γs + fsf (β − α) + gs

g (γ − α)

(s1)

+ (αβγ)13

(αβ βs + α

γ γs + fsf (β − α) + gs

g (γ − α)−α(βs + γs)− βfs(α−β)

f − γgs(α−γ)g

(0).

Recall that all of the curvatures are equal at the outer boundary. At the core,α = β and all of the curvatures as well as g extend to even functions. Thus all ofthe boundary terms cancel. The result follows.

References

[1] Acquistapace, P.; Terreni, B. Fully nonlinear parabolic systems. Recent advances in non-

linear elliptic and parabolic problems (Nancy, 1988), 97–111, Pitman Res. Notes Math. Ser.,

208, Longman Sci. Tech., Harlow, 1989.[2] Agol, Ian. Bounds on exceptional Dehn filling. Geom. Topol. 4 (2000), 431–449.[3] Agol, Ian; Storm, Peter A.; Thurston, William P. Lower bounds on volumes of hyper-

bolic Haken 3-manifolds. (With an appendix by Nathan Dunfield.) J. Amer. Math. Soc. 20(2007), no. 4, 1053–1077 (electronic).

[4] Angenent, Sigurd B.; Knopf, Dan. An example of neckpinching for Ricci flow on Sn+1.Math. Res. Lett. 11 (2004), no. 4, 493–518.

[5] Bleiler, Steven A.; Hodgson, Craig D. Spherical space forms and Dehn filling. Topology35 (1996), no. 3, 809–833.

[6] Buckland, John A. Short-time existence of solutions to the cross curvature flow on 3-manifolds. Proc. Amer. Math. Soc. 134 (2006), no. 6, 1803-1807.

[7] Cao, Xiaodong; Ni, Yilong; Saloff-Coste, Laurent. Cross curvature flow on locallyhomogenous three-manifolds. I. Pacific J. Math. 236 (2008), no. 2, 263–281.

[8] Cao, Xiaodong; Saloff-Coste, Laurent. Cross Curvature Flow on Locally HomogeneousThree-manifolds. II arXiv:0805.3380.

[9] Chow, Bennett; Hamilton, Richard S. The cross curvature flow of 3-manifolds withnegative sectional curvature. Turkish J. Math. 28 (2004), no. 1, 1–10.

[10] Farrell, F. Thomas and Ontaneda, Pedro. A caveat on the convergence of the Ricci

flow for pinched negatively curved manifolds. Asian J. Math. 9, no. 3, 401-406, 2005.

Page 21: CROSS CURVATURE FLOW ON A NEGATIVELY CURVED SOLID …jdeblois/SolidTorusXCF.pdf · CROSS CURVATURE FLOW ON A NEGATIVELY CURVED SOLID TORUS JASON DEBLOIS, DAN KNOPF, AND ANDREA YOUNG

XCF ON A NEGATIVELY CURVED SOLID TORUS 21

[11] Farrell, F. Thomas and Ontaneda, Pedro. On the moduli space of negatively curved

metrics of a hyperbolic manifold. arXiv:0805.2635.

[12] Farrell, F. Thomas and Ontaneda, Pedro. On the topology of the space of negativelycurved metrics. arXiV:math/0607367.

[13] Glickenstein, David. Riemannian groupoids and solitons for three-dimensional homoge-

neous Ricci and cross-curvature flows. Int. Math. Res. Not. (2008), no. 12, Art. ID rnn034,49 pp.

[14] Gromov, Misha; Thurston, William P. Pinching constants for hyperbolic manifolds.

Invent. Math. 89 (1987), no. 1, 1–12.[15] Hodgson, Craig D.; Kerckhoff, Steven P. Rigidity of hyperbolic cone-manifolds and

hyperbolic Dehn surgery. J. Differential Geom. 48 (1998), no. 1, 1–59.

[16] Hodgson, Craig D.; Kerckhoff, Steven P. Universal bounds for hyperbolic Dehn surgery.Ann. of Math. (2) 162 (2005), no. 1, 367–421.

[17] Hodgson, Craig D.; Kerckhoff, Steven P. The shape of hyperbolic Dehn surgery space.Geom. Topol. 12 (2008), no. 2, 1033–1090.

[18] Knopf, Dan; Young, Andrea. Asymptotic stability of the cross curvature flow at a hy-

perbolic metric. Proc. Amer. Math. Soc., 137 (2009), no. 2, 699–709.[19] Lackenby, Marc. Word hyperbolic Dehn surgery. Invent. Math. 140 (2000), no. 2, 243–282.

[20] Ma, Li and Chen, Dezhong. Examples for cross curvature flow on 3-manifolds.

Calc. Var. Partial Differential Equations 26, no. 2, 227-243, 2006.[21] Margulis, Gregory A. The isometry of closed manifolds of constant negative curvature

with the same fundamental group. (Russian) Dokl. Akad. Nauk SSSR 192 (1970), 736–737.

[22] Namazi, Jose; Souto, Juan. Heegaard splittings and pseudo-Anosov maps. Preprint.(http://www.math.uchicago.edu/˜juan/pseudo8.pdf)

[23] Neumann, Walter; Zagier, Don. Volumes of hyperbolic three-manifolds, Topology 24

(1985), no. 3, 307–332.[24] Perelman, Grisha. The entropy formula for the Ricci flow and its geometric applications.

arXiv:math.DG/0211159.[25] Perelman, Grisha. Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109.

[26] Ratcliffe, John G. Foundations of hyperbolic manifolds. Graduate Texts in Mathematics,

149. Springer-Verlag, New York, 1994.[27] Thurston, William P. The geometry and topology of 3-manifolds. Mimeographed lecture

notes, 1979.

(Jason Deblois) University of Illinois at Chicago

E-mail address: [email protected]

URL: http://www.math.uic.edu/~jdeblois/

(Dan Knopf) University of Texas at AustinE-mail address: [email protected]

URL: http://www.ma.utexas.edu/users/danknopf

(Andrea Young) University of ArizonaE-mail address: [email protected]

URL: http://math.arizona.edu/~ayoung